PHYSICAL REVIEW FLUIDS 2, 113701 (2017)

Vortex disruption by magnetohydrodynamic feedback

J. Mak,* S. D. Griffiths, and D. W. HughesDepartment of Applied Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom

(Received 10 September 2016; published 22 November 2017)

In an electrically conducting fluid, vortices stretch out a weak, large-scale magneticfield to form strong current sheets on their edges. Associated with these current sheets aremagnetic stresses, which are subsequently released through reconnection, leading to vortexdisruption, and possibly even destruction. This disruption phenomenon is investigated herein the context of two-dimensional, homogeneous, incompressible magnetohydrodynamics.We derive a simple order of magnitude estimate for the magnetic stresses—and thusthe degree of disruption—that depends on the strength of the background magnetic field(measured by the parameter M , a ratio between the Alfvén speed and a typical flow speed)and on the magnetic diffusivity (measured by the magnetic Reynolds number Rm). Theresulting estimate suggests that significant disruption occurs when M2Rm = O(1). Totest our prediction, we analyze direct numerical simulations of vortices generated by thebreakup of unstable shear flows with an initially weak background magnetic field. Usingthe Okubo-Weiss vortex coherence criterion, we introduce a vortex disruption measure, andshow that it is consistent with our predicted scaling, for vortices generated by instabilitiesof both a shear layer and a jet.

DOI: 10.1103/PhysRevFluids.2.113701

I. INTRODUCTION

The interaction of vortices with a magnetic field is a fundamental process in astrophysicalmagnetohydrodynamics (MHD). Such vortices could be generated, for example, by convection [1,2]or by the breakup of unstable shear flows [3–6]. In the absence of magnetic fields, vortices can becoherent, long-lived structures, particularly in two-dimensional or quasi-two-dimensional systems[e.g., [7]]. However, in the presence of a background magnetic field, various studies have shownhow vortices can be disrupted, by which we mean either a reduction in strength or spatial coherence,or completely destroyed [8–16]. Here we show explicitly how this disruption depends on both thefield strength and on the magnetic Reynolds number Rm.

Astrophysical fluid flows are invariably characterized by extremely high values of Rm. Perhapsthe most important consequence of this is that weak large-scale fields can be stretched by theflow to generate strong small-scale fields, with the amplification being some positive power of Rm[17]. Once the small-scale fields are dynamically significant, the resulting evolution is essentiallymagnetohydrodynamic—rather than hydrodynamic—leading to dramatically different characteris-tics, despite the large-scale magnetic field being very weak. Such behavior has been identified in thesuppression of turbulent transport [18–24], in the suppression of jets in β-plane turbulence [25], andin the inhibition of large-scale vortex formation in rapidly rotating convection [26].

The vortex disruption investigated here, which builds on Ref. [27], and can be contrasted withRef. [28], depends on just such high Rm dynamics. Given that many astrophysical flows are rotatingand stratified, such that the vortices are essentially two-dimensional, it is natural to investigate vortexdisruption in the context of two-dimensional MHD. To quantify when a weak large-scale field canbecome dynamically significant, we first construct a scaling argument for a quite general settingwith a single vortex. We first estimate the amplification of the large-scale field due to stretching

*Current address: Department of Physics, University of Oxford, Clarendon Laboratory, Parks Road Oxford,OX1 3PU; julian.c.l.mak@googlemail.com

2469-990X/2017/2(11)/113701(19) 113701-1 ©2017 American Physical Society

J. MAK, S. D. GRIFFITHS, AND D. W. HUGHES

on the edge of the vortex, following the kinematic arguments of Weiss [29], and then estimate thedynamical feedback on the vorticity. This leads to an explicit prediction for vortex disruption interms of the strength of the large-scale field and Rm.

To investigate vortex disruption in more realistic settings it is necessary to perform directnumerical simulations of the governing MHD equations for freely evolving flows. The vorticesmay be imposed via the initial conditions or, alternatively, may emerge from the development ofan instability of a basic state. Here we adopt the latter approach by considering the instability totwo-dimensional perturbations of two representative incompressible planar shear flows—a shearlayer and a jet—with an aligned magnetic field. This typically leads to the generation of a periodicarray of vortices, each of which may be susceptible to disruption by the magnetic field. Any otherpotential disruption mechanism that requires stratification (e.g., convective instability in the vortexcore due to overturning density surfaces [4,5]) or a third spatial dimension (e.g., elliptic instability[30,31] or the magnetoelliptic instability [32,33]) is ruled out in our numerical simulations, therebyisolating the magnetic field as the sole agent of vortex disruption.

Since magnetic dissipation is crucial in determining the eventual strength of the small-scalefields, and hence the efficacy of the vortex disruption mechanism, it is important to consider howthis is implemented numerically. To have well-defined values of Rm, we carry out simulations withexplicit Ohmic dissipation, represented by a Laplacian, with resolution to the dissipative scale. Thisis in contrast to most other studies of vortex disruption in MHD, in which dissipation is performednumerically at the grid scale, with no explicit diffusion operator [8–10,12–15].

The plan of the article is as follows. Section II contains the scaling argument for the disruption ofa single vortex. The mathematical and numerical formulation of the shear flow problem is given inSec. III. Section IV contains the numerical simulations of vortex disruption; a measure of disruptionbased on the Okubo-Weiss criterion [34,35] is introduced and is used to test the main prediction ofSec. II. Armed with the results of the numerical simulations, in Sec. V we examine in detail the scalingarguments outlined in Sec. II. In Sec. VI we look at the implications for the large-scale dynamicsof the shear flow instability, with particular focus on the mean flow. We conclude in Sec. VII andbriefly discuss possible implications of vortex disruption for the transition to turbulence and mixing.

II. THEORETICAL ESTIMATE OF VORTEX DISRUPTION

Following Weiss [29], we consider an idealized configuration in which a single vortex evolveskinematically in an initially uniform magnetic field of strength B0. On an advective timescale, thefluid motion stretches the weak background field on the edges of the vortex until reconnection of fieldlines occurs; subsequently, the magnetic field lines within the vortex reconnect and are expelled tothe edges of the vortex, a phenomenon known as flux expulsion. Weiss was particularly interested indetermining the scalings of the flux-expelled state, as were Moffatt and Kamkar [36]. Here, however,we are interested in the peak field at the point of reconnection at the edges of the vortex (Weiss’s B1).Since the curved field lines have associated with them magnetic stresses directed toward the vortexcenter, then if B1 is sufficiently strong, the induced stresses will be significant, and we might expectvortex disruption. We are then able to provide an estimate for the dependence of disruption on B0

and the magnetic diffusivity η, using an essentially kinematic argument; put another way, the theoryhere estimates when the kinematic approach breaks down and there must be significant dynamicalfeedback. This approach is similar in spirit to that of Galloway and coworkers [37,38], who consideredthe dynamical feedback of flux ropes formed in magnetoconvection. More recently, Gilbert et al.[28] also considered the dynamical feedback on a vortex, in an idealized, quasilinear setting in whichonly the axisymmetric component of the Lorentz force was retained, thus constraining the vortexflow to remain axisymmetric; this distinguishes it from the fully nonlinear problem considered here.

The magnetic field is governed by the induction equation

∂ B∂t

= ∇ × (u × B) + η∇2 B. (1)

113701-2

VORTEX DISRUPTION BY MAGNETOHYDRODYNAMIC FEEDBACK

We suppose that the vortex has characteristic length Lv and velocity Uv . The initial large-scale fieldB0 is amplified to a stronger small-scale field of strength b and with a small characteristic scale �.Flux conservation across the vortex implies that

b� = B0Lv. (2)

Following the physical arguments of Weiss [29], field amplification is arrested when line stretchingand magnetic diffusion are in balance, i.e., when |B · ∇u| ∼ η|∇2 B|, or UvB0/Lv ∼ ηb/�2.Combining this expression with (2) gives

b ∼(

UvLv

η

)1/3

B0. (3)

The same estimate was obtained via alternative means in Ref. [36].At this stage of the evolution, the magnetic stresses resulting from the Lorentz force may be

estimated. We are interested in the magnetic tension, which may be decomposed as

1

μ0B · ∇ B = −|B|2

μ0rc

en + d

ds

( |B|22μ0

)et , (4)

where μ0 is the permeability of free space, rc is the local radius of curvature, s is the arc length, anden and et are, respectively, the local unit vectors normal and tangential to the magnetic field. For afield expelled to the edge of the vortex, |B| ∼ b, rc ∼ Lv , and d/ds ∼ L−1

v ; hence,

1

μ0|B · ∇ B| ∼ b2

μ0Lv

. (5)

To estimate when the magnetic tension will be dynamically important in the evolution of the vortex,we consider the vorticity equation

∂ω

∂t+ u · ∇ω − ω · ∇u = 1

μ0ρ∇ × (B · ∇ B) + ν∇2ω, (6)

where ρ is the constant density of the fluid. The curl of the magnetic tension involves a transversederivative of the tangential component, and using expression (5), thus scales as b2/(μ0�Lv). We maycharacterize the vortex disruption regime as one in which the curl of the magnetic tension competeswith the advection of vorticity, thereby leading to the scaling

1

μ0ρ

b2

�Lv

∼ Uv

Lv

�v, (7)

where �v ∼ Uv/Lv is the vorticity. Note that here we are assuming that disruption of the vortex, ifit occurs, does so on a much faster timescale than that for reaching the flux-expelled state. Usingexpressions (2) and (3), estimate (7) may be written as

B20

/(μ0ρ)

η∼ �v. (8)

Expression (8) is the dimensional estimate for vortex disruption in terms of the characteristicscales of the vortex. Typically, however, there are other velocity and length scales, U0 and L0,that are used to characterize the flow. Retaining B0 as the characteristic field strength, the relevantnondimensional parameters are

M = B0/√

μ0ρ

U0, Rm = U0L0

η, (9)

113701-3

J. MAK, S. D. GRIFFITHS, AND D. W. HUGHES

where M , the ratio of the Alfvén speed to the characteristic flow speed, is a measure of field strength.Expression (8) may thus be written as

M2Rm ∼ �v, (10)

where

�v = Uv/Lv

U0/L0(11)

is the nondimensional magnitude of the vortex. In many settings, Lv and Uv will be comparable withL0 and U0, in which case the nondimensional estimate for vortex disruption becomes

M2Rm ∼ 1. (12)

The scaling provided by expression (12), first given in Ref. [27], will be tested against simulationdata presented in Sec. IV. The arguments concerning length scales, leading to the estimate (8), willbe revisited in detail in Sec. V.

III. MATHEMATICAL AND NUMERICAL FORMULATION

To examine vortex disruption in detail, we consider the evolution of unstable shear flows ofthe form u = U (y)ex , in the presence of a uniform background magnetic field B = B0ex , in two-dimensional incompressible MHD. Since both u and B are divergence-free, they may be expressedin terms of a streamfunction and magnetic potential, defined by

u = (u,v,0) = ∇ × (ψez), B = ∇ × (Aez). (13)

The z components of the vorticity and current are then given by ω = −∇2ψ and μ0j = −∇2A,respectively. On scaling velocity with a representative flow speed U0, length with a characteristicscale L0, time with L0/U0, and magnetic field with B0, the nondimensional governing equationsbecome

∂ω

∂t− J (ψ,ω) − M2J (A,∇2A) = 1

Re∇2ω, (14a)

∂A

∂t− J (ψ,A) = 1

Rm∇2A, (14b)

−∇2ψ = ω, (14c)

where

J (f,g) = ∂f

∂x

∂g

∂y− ∂f

∂y

∂g

∂x(15)

is the Jacobian operator. The nondimensional parameters are M and Rm, as defined in Eq. (9), andthe Reynolds number

Re = U0L0

ν, (16)

where ν is the kinematic viscosity. In nondimensional form, the two flow profiles we shall considerare U (y) = tanh(y) and U (y) = sech2(y), which we shall refer to as the shear layer and the jet,respectively.

We further decompose the variables in terms of a basic state and a perturbation, i.e.,

ψ = �(y) + ψ, A = y + A, ω = −U ′(y) + ω, (17)

113701-4

VORTEX DISRUPTION BY MAGNETOHYDRODYNAMIC FEEDBACK

where a prime denotes differentiation with respect to y. The system of Eqs. (14) then takes theequivalent formulation (after dropping the tildes on the perturbation terms)

∂ω

∂t+ U

∂ω

∂x− U ′′ ∂ψ

∂x− J (ψ,ω) − M2

[−∂∇2A

∂x+ J (A,∇2A)

]= 1

Re∇2ω − 1

ReU ′′′, (18a)

∂A

∂t+ U

∂A

∂x− ∂ψ

∂x− J (ψ,A) = 1

Rm∇2A, (18b)

−∇2ψ = ω. (18c)

We adopt a domain that is periodic in x and bounded by impermeable, perfectly conducting, andstress-free walls at y = ±Ly , leading to the boundary conditions

ψ = 0, A = 0, ω = 0 on y = ±Ly. (19)

The total energy (the sum of the kinetic energy Ek and the magnetic energy Em) decays as

d

dt(Ek + Em) = − 1

Re

∫∫ω2 dxdy − 1

Rm

∫∫j 2 dxdy, (20)

where Ek and Em are the domain integrals of |u|2/2 and M2|B|2/2, respectively.The channel length is chosen to be some integer multiple of the most unstable wavelength from

linear theory, i.e., Lx = 2nπ/αc, where αc is the most unstable wave number of the associated profiles(αc = 0.44 for the shear layer and αc = 0.90 for the jet [39]). We take n = 1 for the shear layer andn = 2 for the jet, giving domains of roughly equal size. To trigger the instability, we initialize witha perturbation for which the primary instability at wave number αc has fixed amplitude and phase,with other permitted wave numbers αi having smaller amplitude and random phase. Specifically, wetake

(ω,A) =⎡⎣10−3 cos(αcx) + 10−5

�Nx/3�∑i �=n

γi cos(αix − σiLx)

⎤⎦e−y2

, (21)

where γi and σi are randomly generated numbers in [−1,1], and �·� is the floor function. There isa well-defined linear phase of instability, during which the most unstable eigenfunction naturallyemerges. Simulations were run up to t = 150, allowing for an extended nonlinear phase and thepossibility of vortex disruption; with this choice we find that taking Ly = 10 ensures that finiteboundary effects remain negligible.

We solve the system of Eqs. (18) by a Fourier-Chebyshev pseudospectral method, using thestandard Fourier collocation points in x and Gauss-Lobatto points in y, and employing the appropriateFast Fourier Transforms. A semi-implicit treatment in time is employed, treating the dissipation termsimplicitly and the nonlinear terms explicitly. Time-stepping is performed by a third-order accurate,variable time-step, Adams-Bashforth/Backward-Difference scheme, with the step size set by themaximum time step allowed for a fixed CFL number (here taken to be 0.2, which is near marginalfor numerical stability). The equations are solved in spectral space using a fast Helmholtz solver[40], and all runs are dealiazed using Orszag’s 2/3-rule [41] (see Ref. [42] or [43] for further detailsabout the numerical methods employed).

Our simulations are run-down experiments for the evolution of instabilities on a decayingbackground state. To alleviate diffusive effects before the perturbations reach finite amplitude, weremove the diffusion of the basic state (the U ′′′ term in Eq. (18a)) until the perturbation is sufficientlylarge (here measured by the energy possessed by the kx �= 0 Fourier modes). Even so, we foundit necessary to take Re � 500 to produce runs that are not too diffusive and that are qualitativelysimilar to the runs at higher Re. Since the regime estimate (12) naturally suggests a dependence onM and Rm, we fix Re = 500 and vary the other two parameters in the bulk of this work. The required

113701-5

J. MAK, S. D. GRIFFITHS, AND D. W. HUGHES

spatial resolution depends on Rm: the number of x gridpoints Nx and y gridpoints Ny were taken tobe Nx×Ny = 512×1024 for Rm = 1000, 384×768 for Rm = 750, and 256×512 otherwise.

IV. VORTEX DISRUPTION

In this section, we describe the results of direct numerical simulations of freely evolving vorticesgenerated by shear instabilities with a background magnetic field. To measure the disruption of thevortices, we follow Okubo [34] and Weiss [35] in considering the quantity W (x,y,t), defined by

W =(

∂u

∂x− ∂v

∂y

)2

+(

∂v

∂x+ ∂u

∂y

)2

−(

∂v

∂x− ∂u

∂y

)2

. (22)

The bracketed terms are, respectively, the normal and shear components of the rate of strain, andthe vorticity; W thus effectively measures the relative dominance of the strain over the vorticity. Avortex is defined as a region in which W is sufficiently negative. For example, a popular approach isto calculate the standard deviation σ of W and to classify vortical regions by W < −0.2σ . Thoughby no means the only way to identify a vortex [e.g., [44–46]], it is one of the simpler measures thathave been employed previously in a geophysical setting [e.g., [47,48]].

Here we are interested in measuring vortex disruption relative to the purely hydrodynamicevolution. We thus introduce a vortex disruption parameter �(t), defined by

� = 1 −∫A

W dx dy∫A

WHD dx dy, (23)

where WHD is the value of W for the hydrodynamic case. The area A is some portion of physicalspace where there is deemed to be a vortex, as discussed in more detail below. When � = 0, thevortex is not disrupted, whereas � = 1 implies total disruption.

A. Hyperbolic-tangent shear layer

We first consider the disruption of vortices arising from the instability of the shear layer U (y) =tanh(y). Linear instabilities of this profile are well documented for both the hydrodynamic [39]and MHD cases, where linear stability is guaranteed when M � 1 [49,50]. The nonlinear MHDevolution has been studied by numerous authors [e.g., [8–10,12,14,16]]. To provide a good coverageof (M,Rm) space, a total of fifty simulations with parameter values given in Table I were performed,at five values of Rm and ten values of M , where the values of M were chosen to lie well below thestabilising value of M (M ≈ 0.8 for α = 0.44). The initial magnetic to kinetic energy ratio is givenby 2M2Ly/(

∫U (y)2 dy) ≈ (10/9)M2; this ratio is less than 1/90 for the values of M considered

here.We consider a domain supporting a single wavelength of the optimum linear instability mode.

Figure 1 shows snapshots of the vorticity and magnetic field lines from three control runs atRm = 500. These display the representative behaviors for three dynamical regimes: undisturbed(M = 0.01), mildly disrupted (M = 0.03), and severely disrupted (M = 0.05). For M = 0.01, thevorticity is of one sign, and the shear layer rolls up into a vortex. The magnetic stresses are clearly notstrong enough to alter the macrodynamics in any significant way. The vortex evolution is essentiallyhydrodynamic [7], accompanied by magnetic flux expulsion from the vortex. For M = 0.03, we

TABLE I. Parameter values employed for the shear layer simulations.

Parameter Values Marker/color in Fig. 4

M 0.01–0.1, 0.01 spacing + ◦ ∗ × � � , in ascending orderRm 50, 250, 500, 750, 1000 black, red, green, blue, magenta

113701-6

VORTEX DISRUPTION BY MAGNETOHYDRODYNAMIC FEEDBACK

M=

0.01

t = 50(a)M

=0.

01

(b)

t = 75 t = 100 t = 150M

=0.

03M

=0.

03M

=0.

05M

=0.

05

−1

0

1

FIG. 1. Snapshots of (a) vorticity and (b) magnetic field lines for the shear layer at different field strengths(for Rm = Re = 500), shown for the central half of the channel (−Ly/2 � y � Ly/2).

observe the formation of regions of positive vorticity. The magnetic field is no longer confinedto kinematic boundary layers, and the resulting stresses are strong enough to modify the resultingevolution to a certain extent. That said, the vortex is only mildly disrupted and maintains its integrity;there is only a slight decrease of vortex size by the end of the simulation at t = 150. For M = 0.05,the evolution is radically different to the other two cases, with a significant disruption of the vortexand an unconfined magnetic field. By the end of the simulation, only small remnants of the parentvortex persist; vorticity filaments and a complex magnetic field are now the dominant features in thedomain.

Vortex disruption also has a signature in the time evolution of the kinetic and magnetic energies.Shown in Fig. 2 are time series for the three control runs of the mean kinetic energy Ek and meanmagnetic energy Em (defined as the energy content in the kx = 0 Fourier mode), along with theperturbation energies E′

k and E′m (defined as the energy content in the remaining Fourier modes).

The evolution is similar up to t ≈ 60 (cf. Fig. 1), at which time the field amplification is close tobeing arrested by diffusion; the scalings (2) and (3) then apply for the small-scale field, implyingE′

m ∼ b2lLv and Em ∼ B20L2

0, so that E′m/Em ∼ Rm1/3 ≈ 8 here, consistent with Fig. 2. However,

for t � 80, vortex disruption (if it occurs) changes the evolution of the energy. For the undisruptedcase (M = 0.01), the evolution becomes one of complete flux expulsion (see Fig. 1), with E′

m

decreasing to less than Em. (This is different to the well-known theory of Ref. [29], in whichE′

m ∼ Rm1/2Em in the flux-expelled state, but that kinematic single-vortex theory may not applyto this dynamic regime with a periodic array of vortices and remote boundaries.) For the stronglydisrupted case (M = 0.05), we enter a different regime, with E′

m staying close to Em throughout theevolution. This regime with persistent small spatial scales results in stronger dissipation: whereasthe total dissipation is small and comparable with that of the hydrodynamic case for M = 0.01 and0.03, it is about three times higher when M = 0.05. Further, even though Em � Ek throughout the

113701-7

J. MAK, S. D. GRIFFITHS, AND D. W. HUGHES

(a)

(b)

(c)

FIG. 2. Time series of the energies (blue = kinetic; red = magnetic; solid = perturbation state; dashed =mean state; gray solid line = total energy) for the shear layer at different field strengths (for Rm = Re = 500).The curve for mean kinetic energy largely lies on top of the curve for the total energy.

evolution for all three cases (i.e., weak large-scale magnetic field), E′m becomes comparable with E′

k

when there is vortex disruption, reflecting the dynamical importance of the small-scale magnetic field.To calculate the vortex disruption parameter �, defined by Eq. (23), we first need to evaluate

W (x,y,t). This is shown in Fig. 3(a) for the three control runs at t = 150, highlighting those regionswhere either strain or vorticity dominates. Adopting the convention of identifying vortical regionsas those where W < −0.2σ results in the plots of Fig. 3(b). For M = 0.01 and M = 0.03, this leadsto the identification of a well-defined coherent vortex. For the severely disrupted case of M = 0.05on the other hand, disruption results in a small parent vortex, as well as disconnected vortices andfilaments. Our interest is in the disruption to the parent vortex, and so to employ the disruptionmeasure �, we need to ignore these resulting byproducts. To address this, a further filter is applied,which selects the largest connected region originating from the center of the parent vortex (whichin this case is well defined since the resulting instability has zero phase speed, so the single vortexformed is stationary). The W field is digitized, with all points where W < −0.2σ set to one, andall other points set to zero. The MATLAB command bwlabel is then applied to the resulting dataarray to select the connected regions; the connected region originating from the center of the vortexis then chosen as the region A for the calculation of � (see Fig. 3(c)).

The quantity � is computed at t = 150 for all simulations detailed in Table I. To test the vortexprediction (12), in Fig. 4 we show � versus M2Rm, from which it can be seen that there is areassuring collapse of the data. The most important point to note is that for M2Rm � 1.5, thevortices are deemed to have been completely disrupted. This is in agreement with (12), whichpredicts disruption for M2Rm ∼ 1. Further, for M2Rm � 1 we would expect � to be monotonically

113701-8

VORTEX DISRUPTION BY MAGNETOHYDRODYNAMIC FEEDBACK

(a)M

=0.

01(b) (c)

M=

0.03

M=

0.05

−0.8

0

0.8

FIG. 3. Okubo-Weiss field for the shear layer corresponding to the t = 150 cases shown in Fig. 1. (a) Thefull W field given in Eq. (22); (b) the field keeping only W < −0.2σ ; (c) a further filtered field keeping onlythe largest connected region originating from the center of the parent vortex.

increasing with M2Rm. Indeed, on performing a regression on the data in the interval 0.1 � � � 0.9,we find that � ∼ (M2Rm)

1.10(although a different definition of � could lead to a different positive

exponent).Our focus here has been on the transition from an essentially kinematic regime to a dynamic

regime when M2 ∼ Rm−1 � 1. If M2 is increased beyond Rm−1, then severe vortex disruptioncontinues for a while, as implied by Fig. 4. However, if M becomes sufficiently large (M � 0.35here), then the magnetic field is strong enough to suppress vortex formation completely.

B. Bickley jet

We now consider the disruption of vortices arising from the instability of the Bickley jet, U (y) =sech2(y). Linear instabilities of this profile in the hydrodynamic setting are again well documented[39]; there are odd and even modes of instability, with the latter being the most unstable at wave

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

M 2Rm

Δ

FIG. 4. � vs. M2Rm for the shear layer (colors, varying Rm; markers, varying M; see Table I for markerand color values). The dotted line is obtained from a regression of the data. For display purposes, data beyondM2Rm = 5 are omitted.

113701-9

J. MAK, S. D. GRIFFITHS, AND D. W. HUGHES

TABLE II. Parameter values employed for the jet simulations.

Parameter Values Marker/color in Fig. 8

M 0.005–0.05, 0.005 spacing + ◦ ∗ × � � , in ascending orderRm 50, 250, 500, 750, 1000 black, red, green, blue, magenta

number αc = 0.90. In the MHD setting, linear stability is guaranteed for this configuration whenM � 0.5 [50]. This jet profile is known to break up into vortices if the initial field is not so strongthat it suppresses the primary hydrodynamic instability; depending on the parameter values, theresulting vortices have been observed to suffer disruption by the magnetic field [11,13,15].

The optimum wave number αc = 0.90 is roughly twice that of the shear layer case. To employthe same resolution as the shear layer case, we consider a channel length that is twice this optimumwavelength. Fifty simulations with parameter values given in Table II were carried out, with thevalues of M again chosen to lie well below the stabilising value of M (M ≈ 0.3 for α = 0.90,implying lower values of M than for the shear layer). For the Bickley jet, the initial magnetic tokinetic energy ratio is given by 2M2Ly/(

∫U (y)2 dy) ≈ 15M2; this ratio is less than 3/80 for the

values of M considered here.Three control runs are again chosen, with Rm = 500 and M = 0.005, 0.015, and 0.025. Figure 5

shows snapshots of the vorticity and magnetic field lines for the three representative cases. ForM = 0.005, the evolution is essentially hydrodynamic, with a meandering of the jet before it breaksinto two pairs of vortices; MHD feedback is weak and there are no visible disruptions to the vortices.For M = 0.015, the vortices at t = 100 are slightly distorted by the released magnetic stresses;disruption, however, is not strong enough to destroy the vortices. For M = 0.025, the induced

M=

0.00

5

t = 50(a)

M=

0.00

5

(b)

t = 75 t = 100 t = 150

M=

0.01

5M

=0.

015

M=

0.02

5M

=0.

025

−0.7

0

0.7

FIG. 5. Snapshots of (a) vorticity and (b) magnetic field lines for the jet at different field strengths (forRm = Re = 500), shown for the central half of the channel (−Ly/2 � y � Ly/2).

113701-10

VORTEX DISRUPTION BY MAGNETOHYDRODYNAMIC FEEDBACK

0 50 100 15010

−4

10−2

100

102

growth 0.151

0 50 100 15010

−4

10−2

100

102

E

growth 0.151

0 50 100 15010

−4

10−2

100

102

t

growth 0.150

(a)

(b)

(c)

FIG. 6. Time series of the energies (blue = kinetic; red = magnetic; solid = perturbation state; dashed =mean state; gray solid line = total energy) for the jet at different field strengths (for Rm = Re = 500).

magnetic stresses are strong enough to distort the vortices significantly; indeed, the vortex coreshave almost disappeared by t = 150.

Figure 6 shows the time series of the energies for the three control runs. The relative sizes of Ek ,E′

k , Em, and E′m follow the same patterns as for the shear layer. However, in contrast to the shear

layer, here the total dissipation is actually lower for the severe vortex disruption case (M = 0.025)than for the undisrupted case (M = 0.005). As shown in Fig. 5, although the small-scale structuresin the vorticity and magnetic field are initially amplified by severe disruption (t = 100), whichenhances the dissipation, for longer times they are suppressed (t = 150).

We calculate � using a similar procedure as for the shear layer. However, since four vortices aregenerated by the instability in this configuration, to calculate the area A for the integral Eq. (23)we now select the four largest connected components of W (x,y), accounting for periodicity. Theprocedure is illustrated for the three control runs in Fig. 7, at t = 150. When performed for all 50simulations given in Table II, we obtain the plot of � (at t = 150) versus M2Rm given in Fig. 8.It shows the same general characteristics as for the shear layer: there is an approximately linearincrease of � with M2Rm (� ∼ (M2Rm)0.94) up to a critical value, given by M2Rm ≈ 0.3, abovewhich there is complete vortex disruption (� ≈ 1). Note that for both the shear layer and the jet, thecritical value of M2Rm for complete vortex disruption is of order unity, although the precise valuevaries from case to case.

V. SCALING OF THE LORENTZ FORCE

The derivation of expression (12), the criterion for vortex disruption, requires estimates of thestrength and spatial scale of the expelled magnetic field, of the magnitude of the associated Lorentz

113701-11

J. MAK, S. D. GRIFFITHS, AND D. W. HUGHES

(a)M

=0.

005

(b) (c)M

=0.

015

M=

0.02

5

−0.1

0

0.1

FIG. 7. Okubo-Weiss field for the jet corresponding to the t = 150 cases shown in Fig. 5. (a) The full W

field given in Eq. (22); (b) the field keeping only W < −0.2σ ; (c) a further filtered field keeping only the fourlargest connected components, accounting for periodicity. The color scale is saturated to show the small valueswhen M = 0.025.

force and of the competing hydrodynamical forces. In Sec. II, our argument was presented in termsof force balances in the three-dimensional vorticity and induction equations. The phenomena offlux expulsion and subsequent vortex disruption are, however, primarily two-dimensional—and ournumerical simulations are strictly two-dimensional. It is therefore instructive to revisit the scalingarguments in terms of the nondimensional variables ψ and A, as employed in Sec. III, and in thelight of the results of the numerical simulations, described in Sec. IV.

From the nondimensional vorticity Eq. (14a) (or, alternatively, Eq. (18a)), it is clear that themagnetic field becomes dynamically significant once M2J (A,∇2A) is comparable in magnitude toJ (ψ,ω). Our arguments in Sec. II focus on what might be considered as the dynamical breakdownof the kinematic regime. At this point, the velocity and vorticity are still large scale, leading toan unambiguous estimate of the magnitude of J (ψ,ω). By contrast, the expelled magnetic field

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

M 2Rm

Δ

FIG. 8. � vs. M2Rm for the jet (colors, varying Rm; markers, varying M; see Table II for marker and colorvalues). The dotted line is obtained from a regression of the data. For display purposes, data beyond M2Rm = 1are omitted.

113701-12

VORTEX DISRUPTION BY MAGNETOHYDRODYNAMIC FEEDBACK

(∂A/∂x)(∂∇2A/∂y) (∂A/∂y)(∂∇2A/∂x)

−46 0 46

J(A,∇2A)

−1 0 1

FIG. 9. Decomposition of the components in J (A,∇2A), for the shear layer simulation M = 0.05,Rm = 500, at t = 55 [cf. Fig. 1(a)], shown for the central half of the channel (−Ly/2 � y � Ly/2).

varies on both large and small scales; determining the size of the J (A,∇2A) term is thus not sostraightforward.

As explained in Sec. II, flux expulsion leads to magnetic field being confined to a thin strip ofwidth l, along which there are variations in strength on a lengthscale Lv � l. We therefore need tounderstand how the Jacobian

J (A,∇2A) = ∂(A,∇2A)

∂(x,y)= ∂A

∂x

∂

∂y∇2A − ∂A

∂y

∂

∂x∇2A (24)

scales with l and Lv . Unless the field is strictly aligned with one of the x and y axes, then each ofthe two terms on the right-hand side of Eq. (24) will scale as A2/l4. One might therefore be temptedto assume that

J (A,∇2A) ∼ A2

l4; (25)

however, this significantly overestimates the magnitude of the Jacobian. To see why this is the case,it is instructive to express J (A,∇2A) in general two-dimensional orthogonal curvilinear coordinates(x1,x2) with scale factors (h1,h2), namely

J (A,∇2A) = 1

h1h2

(∂A

∂x1

∂∇2A

∂x2− ∂A

∂x2

∂∇2A

∂x1

), with

∇2A = 1

h1h2

(∂

∂x1

(h2

h1

∂A

∂x1

)+ ∂

∂x2

(h1

h2

∂A

∂x2

)). (26)

Now consider an arbitrary point on the (curved) flux strip, and let (x1,x2) be a local orthogonalcurvilinear coordinate system in the plane of the strip, with x1 and x2 pointing, respectively, acrossand along the strip. Then, with h−1

1 ∂/∂x1 ∼ l−1 and h−12 ∂/∂x2 ∼ L−1

v , we have ∇2A ∼ A/l2, and

J (A,∇2A) ∼ A2

l3Lv

, (27)

a reduction of O(l/Lv) in comparison with the naive estimate (25).To test our theoretical predictions quantitatively, we use numerical simulations of the shear layer

at a time when strong field has formed at the edge of the vortex, but before disruption occurs. We havechosen t = 55 (cf. Fig. 1), although the results are a little sensitive to this choice. Confirmation that,in Cartesian coordinates, significant cancellation between the two component terms of J (A,∇2A)does indeed take place is provided by Fig. 9, which shows, individually, the magnitudes of the twoterms in Eq. (24) for one specific case, together with their much smaller difference.

To quantify this reduction across a wide parameter space, we evaluate

δ = max |J (A,∇2A)|max |(∂A/∂x)(∂∇2A/∂y)| . (28)

113701-13

J. MAK, S. D. GRIFFITHS, AND D. W. HUGHES

0 0.02 0.04 0.060

0.002

0.004

0.006

0.008

0.01

0.012

∼ ( Lv)0.930

Lv

δ

(a)

0 250 500 750 10000

0.2

0.4

0.6

0.8

1

∼ Rm1.111

Rm

max

(J(A

,∇2A

))

(b)

FIG. 10. (a) Plot of δ as defined in Eq. (28) against �/Lv , for the t = 55 simulation data of the shear layeracross all values of M and Rm (colors, varying Rm; markers, varying M; see Table I for marker and colorvalues); the Rm = 50 data are not plotted. (b) Plot of the mean value of max J (A,∇2A) against Rm, where theaverage is taken over the ten values of M . Both dotted lines are obtained from a regression of the data with theRm = 50 data omitted.

If J (A,∇2A) ∼ A2/�3Lv , as in (27), but the individual components scale as A2/�4, as in (25), weshould have a reduction

δ ∼ �

Lv

. (29)

For configurations such as shown in Fig. 9 we can independently determine δ, �, and Lv . The lengthscale � is diagnosed by taking a transect at y = 0 of |(∂A/∂x)(∂∇2A/∂y)| and calculating the halfwidth of the single maximum. The length scale Lv is identified as the mean extent of the vortex asidentified by the Okubo-Weiss criterion. Figure 10(a) shows δ as a function of �/Lv for 40 runs withRm � 250. A fitting of the logarithm of this data shows that δ scales as (�/Lv)0.930, consistent withthe prediction in (29). (Note that we have omitted the Rm = 50 points; unsurprisingly, for such alow value of Rm, they do not obey the same relation between δ and �/Lv .)

We are ultimately interested in how J (A,∇2A) at this time scales with Rm, so we now test thescaling (27) across a wide parameter space. Note that A and Lv are of order unity by the assumednondimensionalization, while the nondimensional versions of (2) and (3) are b� ∼ 1 and b ∼ Rm1/3,so that � ∼ Rm−1/3. Our scaling (27) thus implies J (A,∇2A) ∼ Rm at this time. Figure 10(b) showsthe mean value of max J (A,∇2A), where the average is taken over all ten values of M at eachRm (see Table I). Omitting the Rm = 50 data, a fitting of the logarithm of this data shows thatmax J (A,∇2A) scales as Rm1.111, consistent with the prediction in (27).

The ordering (27) is essentially equivalent to the ordering (5), although the details of the derivationare slightly different; it thus leads to the estimate (12) for vortex disruption, namely M2Rm ∼ 1. Werethe Lorentz force stronger than estimated by (27), in particular, if its strength were given by (25),then a weaker background field would lead to vortex disruption, determined by M2Rm4/3 ∼ 1.Conversely, if the cancellation in expression (24) were extremely strong (i.e., O(l2/L2

v)), leadingto an ordering J (A,∇2A) ∼ A2/l2L2

v , then the threshold for the dynamic regime would satisfyM2Rm2/3 ∼ 1. It is of interest to note that it is this latter scaling that is identified by Gilbert et al.[28] as the onset of the dynamic regime in their quasi-linear model of flux expulsion.

113701-14

VORTEX DISRUPTION BY MAGNETOHYDRODYNAMIC FEEDBACK

VI. DYNAMICAL PHENOMENA

We believe that the dynamics underlying the vortex disruption, leading to the estimate (12),is quite general. However, in any given setting, there could be interesting dynamical implicationsbeyond those of the disruption itself. Here, we spell out some of the specific implications for thenonlinear evolution of shear flow instabilities.

A. Mean flow changes

A quantity of considerable importance in the instability of shear flows is the mean flow, definedhere to be

u(y,t) = 1

Lx

∫ Lx

0u(x,y,t) dx, (30)

where Lx is the channel length. In combination with the magnetic field, the mean flow determinesthe strength of the initial linear instability, and its evolution with time shows how the nonlineardynamics act to mix momentum in the cross-stream direction.

Snapshots of u for the shear layer and the jet are shown in Fig. 11, for each of the three controlruns. For the undisturbed cases (essentially hydrodynamic) in Figs. 11(a) and 11(d), the shear isreduced around the center of the channel as the instability reaches finite amplitude, but the meanflow remains largely unchanged thereafter. For the mildly disrupted cases in Figs. 11(b) and 11(e),the behavior is similar (i.e., essentially hydrodynamic) near the center of the channel (|y| � 2), butbeyond that there is an additional broadening of the mean flow when vortex disruption sets in, fort � 100. However, for the severely disrupted cases in Figs. 11(c) and 11(f), the mean flow evolutionis substantially different, with mixing of momentum over a wider region (about twice as wide asfor the hydrodynamic case), leaving smaller shears near the center of the channel. This is consistentwith previous studies [e.g., [16]].

It is possible to quantify the influence of vortex disruption on the evolving mean flow by measuringthe width of the mean flow changes relative to those of the hydrodynamic case [27]. This analysis

−1 0 1−10

0

10

y

(a) M = 0.01

−1 0 1−10

0

10

u

(b) M = 0.03

−1 0 1−10

0

10(c) M = 0.05

t = 0t = 50t = 75t = 100t = 150

0 0.5 1−10

0

10

y

(d) M = 0.005

0 0.5 1−10

0

10

u

(e) M = 0.015

0 0.5 1−10

0

10(f) M = 0.025

FIG. 11. Snapshots of u for the shear layer (a, b, c) and jet (d, e, f) at different field strengths (forRm = Re = 500). Some of the curves lie on top of each other and are indistinguishable.

113701-15

J. MAK, S. D. GRIFFITHS, AND D. W. HUGHES

reveals how the width of the changes to the mean flow increases with M2Rm, with substantialwidening when M2Rm ∼ 1—i.e., within the vortex disruption regime.

B. Secondary hydrodynamic instabilities

It is well known that vortices generated during the finite-amplitude stage of shear instabilitiescan be unstable to a range of secondary hydrodynamic instabilities. Our numerical simulations weredesigned to suppress such secondary hydrodynamic instabilities, so that only magnetic disruption ofthe vortices could occur. However, within our two-dimensional system, there is the possibility of thesubharmonic pairing instability [51–53], which requires two or more vortices in the along-streamdirection. For our shear layer simulations, this was completely suppressed, since only a single vortexwas generated within the domain. For our jet simulations, where two vortices were generated inthe along-stream direction, early signatures of the subharmonic pairing instability may be seen inFig. 5 for the undisturbed case with M = 0.005, and to a lesser extent in the mildly disrupted casewith M = 0.015. However, in the strongly disrupted case with M = 0.025, the magnetic disruptionoccurs on a faster timescale than the pairing instability, and the disrupted vortices show no signsof pairing. An interesting alternative scenario is when magnetic disruption does not act on theprimary vortices, but in which repeated subharmonic pairings eventually lead to a large vortex thatdoes suffer magnetic disruption. Although there is evidence of this occurring in studies of longerchannels [14], we did not find this in sample simulations where the domain was extended to allowfor eight wavelengths of the primary instability.

For different flow configurations, other hydrodynamic secondary instabilities could occur. Witha third spatial dimension, there is the possibility of either hyperbolic instabilities on the thin braidsbetween vortices or ellipitical instabilities on the vortex cores, as discussed in Ref. [54], for example.With density stratification, there is also the possibility of convective type instabilities associated withdensity overturning in the vortex cores, which compete with various other modes [4,5]. However,such secondary instabilities are all excluded here through our choice of a two-dimensional system ofconstant density. In three-dimensional or stratified flows, whether or not such secondary instabilitieswould act before magnetic disruption of the parent vortices is an open question.

VII. CONCLUSION AND DISCUSSIONS

Of great astrophysical significance is the idea that a very weak large-scale magnetic field, i.e.,a field with energy much smaller than the kinetic energy of the flow in question, can still havedynamically significant consequences. Small-scale, typically turbulent, motions distort the large-scale field to generate small-scale magnetic fields, amplifying the field strength by some positivepower of the magnetic Reynolds number Rm. Since the defining characteristic of astrophysicalplasmas is that Rm � 1, this implies that weak large-scale magnetic fields cannot simply be ignored.This phenomenon has been explored for the suppression of both turbulent magnetic diffusivity[17,18,23,24] and the turbulent α-effect of mean field electrodynamics [19–22], the inhibition of jetformation in β-plane turbulence [25], and the suppression of large-scale vortices in rapidly rotatingconvection [26]. Here we have explored how, in another broad class of problems—the disruption ofvortices by a weak large-scale field—the same general dynamical processes can occur. By adoptingthe arguments of Ref. [29], we are able to estimate the magnitude of the induced magnetic stressesarising from the stretching of magnetic field lines by the swirling fluid motion, noting that there is anontrivial cancellation in the curl of the magnetic tension, as supported by numerical and theoreticalanalyses. Vortices are disrupted when these magnetic stresses released are sufficiently large; thisoccurs when M2Rm ∼ 1, where M2 measures the energy of the large-scale field relative to thekinetic energy. Thus, when Rm � 1, vortex disruption occurs for M � 1.

The estimate for vortex disruption, M2Rm ∼ 1, is widely applicable for vortex dynamics, howeverthe vortices may arise. Here we test the criterion in detail by considering one of the most importantmeans of vortex generation—the nonlinear development of shear instabilities. We focus on two-dimensional, homogeneous, incompressible MHD, considering in detail two prototypical shear

113701-16

VORTEX DISRUPTION BY MAGNETOHYDRODYNAMIC FEEDBACK

flows (the hyperbolic tangent shear layer and the Bickley jet) with a weak background magneticfield. The instability of both these flows leads to a periodic array of vortices, which can be proneto magnetic disruption. We first identify coherent vortices by the Okubo-Weiss procedure, whichprovides a simple definition of vortices as those regions where vorticity dominates over strain, andthen construct a measure of disruption � by comparison with the purely hydrodynamic evolution.By performing fifty simulations to cover a range of M (0.01 � M � 0.1 for the shear layer and0.005 � M � 0.05 for the jet) and Rm (50 � Rm � 1000) at Re = 500, we are able to investigatethe dependence of � on both M and Rm. For both shear flows, we find an approximate linear increaseof � with M2Rm up to a critical value of M2Rm of order unity, followed by a sharp transition toa regime in which � ≈ 1, denoting total disruption of the vortices. These numerical results arein excellent agreement with the theoretical estimate. Our theoretical ideas are also supported byinspection of the energy time series, which show that disruption is characterized by an orderingin which the energies of the perturbed magnetic field, the mean field, and the perturbed flow arecomparable (but all much less than the kinetic energy of the unperturbed shear flows).

The disruption estimate M2Rm ∼ 1, first derived in Ref. [27], is in contrast to the result morerecently reported in Ref. [28]. In that work, which considers the dynamical feedback of a vortexin a magnetic field, in a quasilinear setting assuming azimuthal symmetry, the breakdown of thekinematic argument, and thus the regime where vortex disruption may be expected to take place,occurs when M2Rm2/3 ∼ 1 in our notation. Since our set of simulation data takes Rm values between50 and 1000, distinguishing between the asymptotic scalings M2Rm and M2Rm2/3 is difficult. Thatsaid, it is interesting to note, following our line of argument in Sec. V, that if the curl of the magnetictension were to scale as b2/L2

v , rather than b2/�Lv as we suggest, then the resulting disruptionestimate would indeed be M2Rm2/3 ∼ 1. Given that both our theoretical and numerical analysessupport the scaling of the curl of the magnetic tension as b2/�Lv , the difference in the asymptoticscaling for disruption is perhaps attributable to differences between a fully nonlinear system and aquasilinear system with imposed symmetry.

Further, the vortex disruption estimate depends crucially on a balance between advection anddissipation of small-scale field. Since dissipation is represented by a Laplacian operator in theinduction equation, estimate (12) involves an explicit dependence on Rm. This dependence iscaptured by our numerical approach, which uses a Laplacian diffusion operator with resolutiondown to the dissipation scale. Numerical schemes with alternative (non-Laplacian) prescriptions forthe dissipation would presumably realize vortex disruption in a somewhat different way.

The notion of vortex disruption has some interesting astrophysical implications. Since the derivedestimate for disruption is M2Rm ∼ 1, and Rm is typically extremely large in astrophysical systems,vortex disruption should be a robust dynamical feature. If vortex disruption does occur, it is likely tolead to mixing of quantities such as angular momentum, heat and passive scalars, with implicationsfor the large-scale angular velocity, temperature and chemical composition. Of particular note is thatin systems such as the solar tachocline, constrained by stable stratification, many of the standardmixing scenarios do not occur. However, provided there are vortices, we have shown that theirinteraction with a magnetic field can provide an alternative route to mixing. The theoretical disruptionestimate (12) assumes that Rm � 1 and implicitly assumes that the flow is smooth on the small scalesof the field; this equates to an assumption that the magnetic Prandtl number Pm = ν/η = Rm/Re �O(1). Whereas this does indeed hold in the interstellar medium, in stellar interiors Pm � 1. Althoughone could envisage vortex disruption by a similar physical mechanism in this regime, the line ofargument leading to a disruption estimate would need to be modified. Furthermore, the testing ofany such hypothesis in the regime Re � Rm � 1 is currently computationally unattainable.

ACKNOWLEDGMENTS

J.M. was supported by the STFC Doctoral Training Grant No. ST/F006934/1, the ISF Grant No.1537/12, and the NERC Grant No. NE/L005166/1. We thank Andrew Gilbert for helpful comments,which led to further analysis and a subsequent improvement of this article.

113701-17

J. MAK, S. D. GRIFFITHS, AND D. W. HUGHES

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